Global Homeomorphism and Covering Projections on Metric Spaces

TL;DR

This paper establishes sufficient conditions under which local homeomorphisms on certain metric spaces become covering projections, extending classical results like Hadamard's theorem and exploring implications for global implicit functions.

Contribution

It introduces new path continuation criteria for covering projections on metric spaces with local structure, extending classical theorems and analyzing quasi-isometric mappings.

Findings

Sufficient conditions for local homeomorphisms to be covering projections.

Extension of Hadamard's global inversion theorem in metric spaces.

Applications to the existence of global implicit functions.

Abstract

For a large class of metric spaces with nice local structure, which includes Banach-Finsler manifolds and geodesic spaces of curvature bounded above, we give sufficient conditions for a local homeomorphism to be a covering projection. We first obtain a general condition in terms of a path continuation property. As a consequence, we deduce several conditions in terms of path-liftings involving a generalized derivative, and in particular we obtain an extension of Hadamard global inversion theorem in this context. Next we prove that, in the case of quasi-isometric mappings, some of there sufficient conditions are also necessary. Finally, we give some applications to the existence of global implicit functions.